One of the main difficulties in estimating the \(R_0(a)\)  function is its non-identifiability in general. Recent studies [1]}, [2]}, [3]}, [4]}, [5]} developed methods to identify the various parameters for the COVID-19 pandemic by using cumulative reported cases data and differential equations models. Differential equations can be written in the form studied here by assuming that \(R_0(a)\)  (or equivalently, \(\beta (a)\) ) is independent of \(a\) . Suppose that we are restricted to a period when the data is growing exponentially fast. If we take a fixed function \(\beta (a)\) , then by adapting the method developed in [1]}, we could identify a transmission rate \(\tau \)  so that the output of the model stays very close to the data, for any function \(\beta (a)\). The same could be achieved with a good phenomenological description of the data by using the method developed in [2]}, [3]}, [4]} with a time-dependent transmission rate. This means that the reported cases data is not sufficient to determine accurately the function \(R_0(a)\) . Without a good description of the initial distribution, it is hopeless to identify \(R_0(a)\)  by using reported cases only.
